Influence of Armour Porosity on the Hydraulic Stability of Cube Armour Layers

27 Pages·2015·0.53 MB·English

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This paper must be cited as:
Medina Folgado, JR.; Molines Llodra, J.; Gómez Martín, ME. (2014). Influence of armour

porosity on the hydraulic stability of cube armour layers. Ocean Engineering. 88:289-297.
doi:10.1016/j.oceaneng.2014.06.012.

The final publication is available at

http://dx.doi.org/10.1016/j.oceaneng.2014.06.012

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Elsevier
Influence of Armour Porosity on the Hydraulic Stability of
Cube Armour Layers
Josep R. Medina1, Jorge Molines2, and M. Esther Gómez-Martín3
1Professor and 2Asst. Res., Dept. of Transportation, ETSI Caminos, Universitat Politècnica de
València, Camino de Vera s/n, 46022 Valencia, Spain. Tel.: [phone]; Fax: +34
963877379. [email protected] (corresponding author), [email protected]
3Asst. Prof., Dept. of Civil Engineering, Escuela Politécnica Superior, Universidad de
Alicante, San Vicent del Raspeig s/n, 03690 Alicante, Spain. Tel.: [phone]; Fax: +34
965903678. [email protected]

Abstract
Armour placement and packing density directly affect construction costs and hydraulic
performance of mound breakwaters. In this paper, the literature concerning the influence of
armour porosity on the hydraulic stability of single- and double-layer armours is discussed.
Qualitative and quantitative estimations for the influence of armour porosity and packing
density on the hydraulic stability are given for the most common concrete armour units. The
analysis focuses on specific 2D hydraulic stability tests of double-layer cube armours with
different armour porosities and permeable core. The packing density showed a 1.2-power
relationship with the stability number for cube units. The literature review and experimental
results with small-scale breakwater models protected with a variety of armour units clearly
indicate that a significant increase in armour porosity above the recommended values
substantially decreases armour hydraulic stability. To avoid uncontrolled model effects,
packing density should be routinely measured in small-scale tests, and armour placement
techniques should be monitored at prototype scale. The actual packing density obtained in
small-scale models and prototypes has to be explicitly reported, because packing density
significantly affects hydraulic stability during service time.
Key words:
Breakwater, Heterogeneous Packing, Armour porosity, Packing density, Armour
damage, Hydraulic stability, Armour unit, Cubic block.
Highlights:
(1) Armour porosity affects the hydraulic stability of armour layers.
1
(2) Armour porosity or packing density is rarely included in armour stability formulae.
(3) Packing density of randomly-placed double-layer cube armours is a relevant design
factor affecting hydraulic stability and construction costs.
(4) A significant 1.2-power relationship has been found between the stability number
and packing density for double-layer cube armours.
(5) The packing density of the armour layer has to be explicitly reported in small-scale
tests and monitored at prototype scale.

1. Introduction
Rock armoured mound breakwaters have been built for centuries. When breakwaters had to
be constructed in harsher environmental conditions, larger stones were needed for armour
layers. In the 19th century, when local quarries were not able to provide stones of the
appropriate size and price, precast concrete cubes and parallelepiped blocks were introduced
and numerous precast concrete armour units were designed later to optimize the armour layer
of mound breakwaters. The overall breakwater construction cost depends on numerous
factors, these being associated to design and logistic factors, including the type of armour
material (unreinforced concrete, granite rock, sandstone rock, etc.), armour unit mass,
personnel and material unit costs, total concrete consumption, placement equipment, casting,
handling and stacking procedures, etc. This paper focuses on armour porosity and the
associated packing density, because these two parameters significantly affect breakwater
hydraulic performance, construction costs and payments.
Hudson’s formula published in 1959, popularized later by USACE (1984), focused the attention
of the engineering community on the stability coefficients (K ) associated with different armour
D
units, randomly placed in double-layer armours with a prescribed nominal porosity, P, and a
layer coefficient, k . Using the equivalence H=H in the original Hudson formula, Eq. 1 is
Δ s
known as the generalized Hudson formula, still widely used by practitioners to compare different
breakwater designs at the preliminary stage, including double- and single-layer armours,
H3ρ H3ρ
M = s r = s r (1)
ρ 3 ∆3K cotα
D
K  r −1 cotα
D ρ
 
w
in which K is the stability coefficient; M is the armour unit mass; H is the incident significant
D s
wave height; α is the slope angle: Δ=(ρ/ρ -1) is the relative submerged mass density; and ρ and
r w r
2
ρ are the armour unit and water mass densities, respectively. The equivalent cube size or
w
nominal diameter of the armour units is defined as D =(M/ρ)1/3; and Eq. 1 can be re-written as a
n r
function of the stability number, N=H/(ΔD )=(K cotα)1/3.
s s n D
If Eq. 1 is used to compare different armour units in similar storm conditions, the higher the K
D
is, the lower armour unit mass and concrete consumption. Since the invention of the Tetrapod in
1950, numerous armour units have been invented in the search for high values of K and the
D
corresponding economic savings. Usually, high values of K are associated with complex
D
armour unit shapes (e.g. Dolos), reducing concrete consumption and requiring smaller rocks in
the filter layer and lighter placement cranes. These savings should exceed the additional costs
associated with more expensive concrete and complex formworks, production, handling,
stacking and placement compared to the simple and easy to handle conventional cubes and
parallelepiped blocks.
The invention of AccropodeTM in 1980 and other interlocking units later, designed for single-
layer armouring, significantly reduced concrete consumption and cost (see Vincent et al., 1989,
and Holtzhausen, 1998). Structural integrity is a key issue when using these bulky units as are
adequate placement and packing density to guarantee interlocking of units during service time
(see Jensen, 2013, and Latham et al., 2013). In the preliminary design phase, armour porosity
and placement technique are usually considered as secondary factors, which are either explicitly
prescribed (see Mouquet, 2009, and Paulsen and Wareing, 2009) for single-layer armours or
implicitly defined by engineering manuals (e.g. USACE, 1984, and CIRIA et al., 2007) for
randomly-placed double-layer armours. There are other environmental and structural
characteristics not included in Eq.1 which may also have a significant influence on the armour
stability, such as packing density, Iribarren’s number, core permeability and relative crest-
freeboard; this paper focuses attention on packing density because it is frequently a key factor
affecting construction cost, concrete consumption and breakwater safety.
At prototype scale, armour units are usually placed using crawler cranes; armour porosity and
placement below mean water level (MWL) are not easy to control due to poor visibility, waves
and wind (see Medina et al., 2010). Because armour porosity and packing density are not
explicitly included in most of the hydraulic stability formulae used by practitioners, such as Eq.
1, short-term cost optimization tends to increase armour porosity of prototypes above tested and
recommended values. Unfortunately, a significant increase in the armour porosity usually leads
to a significant reduction in hydraulic stability. This paper analyses the influence of packing
density on the hydraulic stability with special attention to double-layer cube armours. The aim is
3
to estimate the model effect associated to armour porosity, which can differ substantially
between the prototype and the corresponding small-scale model tested in laboratory.
Section 2 includes a literature review concerning the effects of armour porosity on the hydraulic
stability of different armour units. Section 3 focuses on porosity changes within cube armours
due to Heterogeneous Packing (HeP) and explains the virtual net method used to measure
armour damage in the small-scale tests reported in this research. Section 4 describes the 2D
hydraulic stability tests of cube models with different armour porosities carried out for this study
including the analysis of the experimental results. Finally, the most relevant conclusions of this
research are provided in Section 5.

2. Literature on the influence of armour porosity on hydraulic stability
Porosity is an intuitive general concept widely used to refer to the volume of voids in a granular
system. Nevertheless, armour porosity is not always easy to determine; armour thickness must be
defined first, which may be an easy task for orderly-placed armour units but not so
straightforward for randomly-placed units. Armour thickness of randomly-placed units is usually
referred to as n=1 (single-layer) or n=2 (double-layer) times the equivalent cube size,
nD =n(M/ρ)1/3. However, most engineering manuals (e.g. USACE, 1984 and CIRIA, 2007)
n r
recommend, for each unit, a specific layer coefficient or layer thickness factor, k , and a specific
Δ
nominal porosity, P, called “fictitious porosity” by Zwamborn (1978). Placing density
(φ[units/m2]) is a real physical variable which is controlled by the placement grid and is related
to k and P according to
Δ
2
ρ  3
ϕ= n(k )(1−P) r  (2)
∆
M 
in which n is the number of layers in the armour; k is the layer coefficient; P is the nominal
Δ
porosity; and M/ρ=D 3 is the volume of the armour unit. Different pairs of k and P lead to the
r n Δ
same placing density, φ; thus, Frens (2007) drew attention to misinterpretations caused by the
use of different criteria regarding the layer coefficient and the porosity concept. In order to
prevent misunderstandings, for randomly-placed armours, this paper refers to the number of
layers n, the packing density φ=φD 2, and armour porosity p=1-φ/n, being the packing density a
n
useful parameter to measure the relative consumption of concrete in the armour layer, which may
be considered the dimensionless placing density:

φ=ϕD2 = n(k )(1−P)= n(1− p) (3)
n ∆
4
in which φ is the placing density; and p is the armour porosity parameter used in this research.
Armour porosity p is equal to nominal porosity P only when considering a layer coefficient of
k =1.00. Therefore, this study uses a two-parameter armour characterization (n and p or φ),
Δ
instead of the conventional three-parameter characterization (n, P and k ). For instance, USACE
Δ
(1984) recommended n=2, P=50% and k =1.04 for double-layer Tetrapod armours, which is
Δ
equivalent to n=2 and p=48% or φ=1.04. Using different notations for the same concepts,
CIRIA (2007) recommended n=2, P=50% and k =1.02 for Tetrapod armours which is equivalent
Δ
to n=2 and p=49% or φ=1.02.
Armour porosity and placement can be very well controlled in small-scale tests; dry
construction, perfect view of the armour layer and placement by hand are ideal laboratory
construction conditions which do not exist at prototype scale. On the contrary, prototype
conditions usually involve placement grids, crawler cranes, blind underwater placement and
other restrictions that generate uncertain armour porosities and these may significantly change
in space and time (see Medina et al., 2010, and Latham et al., 2013). In this paper, the
literature review is focused on armour porosity and its influence on hydraulic stability. The
analysis of the literature reveals that a significant reduction in the packing density φ below
the recommended value results in a significant decrease in the hydraulic stability of the
armour.

2.1 Single- and double-layer rock armours
Hald et al. (1998) carried out small-scale tests of single-layer rock armours with cotα=1.5 and
different placement techniques, which may be used to describe the hydraulic performance of
hundreds of single-layer rock armours in rubble-mound breakwaters built in Norway since
1886. Armour porosity p=40% [φ=0.60] was assumed to compare armour damage measured
with profiles and visual counting techniques, and results were compared with conventional
double-layer rock armours. Single-layer rock armours with orderly-placed rocks were more
stable than double-layer rock armours with randomly-placed rocks, which were much more
stable than single-layer rock armours with randomly-placed rocks. Vandenbosch et al. (2002)
conducted small-scale tests of single-layer rock armours with slopes in the range
1.5≤cotα≤3.0; different placing densities were obtained varying the layer thickness and
placement techniques. Armour porosity p=30% (φ=0.70) was considered “normal” and a
decrease in packing density resulted in decreased armour stability. USACE (1984)
5
recommended p=37% (φ=1.26) for double-layer rough and smooth quarrystone armours;
however, armour porosity is not easy to measure. Latham et al. (2002) provided a rapid
survey method to estimate packing densities and analysed 23 real breakwaters with
30.0%≤p≤40.1% having an average porosity of rock armour p=34% (φ=1.32).
The literature on rock armour described above clearly indicates that randomly-placed rock
armours show higher hydraulic stability if packing density is higher. Double-layers are more
stable than single-layer armours, and hydraulic stability decreases if armour porosity
increases. On the other hand, the study by Hald et al. (1998) on Norwegian breakwaters
proved that orderly-placed single-layer rock armours with high packing densities have higher
hydraulic stabilities than conventional randomly-placed double-layer rock armours.

2.2 Single- and double-layer Tetrapod armours
Van der Meer (1999) analysed small-scale test results of randomly-placed single- and double-
layer Tetrapod armours corresponding to models having cotα=1.5 and a packing density
0.48≤φ≤1.02; for surging waves, the stability number was explicitly related to packing
density according to:
H  N0.5   φ 
Ns = ∆Dsn =3.75N0o.d25 +0.850.40+0.61φSPM som−0.1 (4)
and the formula for plunging waves resulted in:
H  N0.5   φ 
Ns = ∆Dsn =8.6 N0o.d25 +3.940.40+0.61φSPM som−0.1 (5)
in which N is the stability number; s =H /[gT 2/2π] is the wave steepness using the mean
s om s m
period; N is the number of waves; φ is the packing density; φ =1.04 (n=2 and p=48%) is
SPM
the recommended packing density given by USACE (1984); and N is the relative damage
od
level which may be comparable to half the dimensionless damage, S (see Van der Meer,
1999). Regardless the number of layers (n=1 or 2), the higher the packing density of Tetrapod
armours, the higher the hydraulic stability will be.
Vandenbosch et al. (2002) conducted small-scale tests of single-layer Tetrapod armours with
slope cotα=1.5 and packing densities φ=0.78, 0.73 and 0.62, and later De Jong et al. (2004)
tested similar single-layer Tetrapod models; these experimental results confirmed that higher
packing densities generated higher hydraulic stability.
Gürer et al. (2005) carried out small-scale tests of double-layer Tetrapod armours, using two
6
specific placement techniques, slope cotα=1.5 and packing density φ=0.94 and 0.80; the
placement technique with the higher porosity showed higher hydraulic stability for initial
damage, but initial damage was followed by a rapid failure beyond a critical wave height.
Taking into account the literature reviewed herein, it is clear that single- and double-layer
Tetrapod armours with randomly-placed units have higher hydraulic stability if packing
density is higher; however, special placement techniques and armour patterns may also
significantly influence specific failure functions.

2.3 Single- and double-layer cube armours
Van Gent et al. (1999) tested single-layer cube armours, randomly-placed, with slope
cotα=1.5 and p%=25%, 30% and 40% (φ=0.75, 0.70 and 0.60); hydraulic stability increased
when armour porosity decreased, being p=25% the most stable armour with significant face-
to-face fitting arrangements after wave attack. Van Buchem (2009) carried out similar small-
scale tests of single-layer cube armours, but placed by hand in ordered horizontal rows, with
slopes cotα=1.5 and 2.0 and p= 20%, 28% and 35%; hydraulic stability was the highest for
p=28% and decreased for p=20% and p=35%.
Although basing his experiments only on Tetrapod units, Van der Meer (1999) postulated a
similar function to estimate the influence of armour porosity on the hydraulic stability of cube
armours; the explicit formula proposed for cube armour was:
H  N0.4   φ     R 
Ns = ∆Dsn =6.7 No0d.3 +1.00.40+0.61φSPM som−0.11+0.17exp−0.61Dcn  (6)
with the same notation used in Eqs. 4 and 5 but using φ =1.17 (n=2 and p=42%); R is the
SPM c
crest freeboard; and D =(M/ρ)1/3 is the equivalent cube size or nominal diameter.
n r
USACE (1984) and CIRIA (2007) recommended p=42% for double-layer modified cube and
cube armours with randomly-placed units. Vandenbosch et al. (2002) conducted small-scale
tests of double-layer cube armours, also randomly-placed, but with slope cotα=1.5 and
p=25%, 30% and 40%; hydraulic stability was reported much higher for p=30% than p=40%
but it decreased for p=25%. Nonetheless, it should be considered that when crawler cranes are
used and underwater viewing conditions are poor, it is not possible to randomly place cubes
having p≤35% (see Medina et al., 2010). Therefore, double-layer randomly-placed cube
armour has higher hydraulic stability if packing density is higher, although the placement
technique at prototype scale may then be a critical issue for high packing density armours.
Yagci and Kapdasli (2003) carried out small-scale tests of double-layer Antifer cubes with a
7
modified random technique called “irregular placement”, cotα=2.0 and packing densities
φ=0.94 (p=53%) and φ=1.13 (p=43.5%). Yagci et al. (2004) later tested double-layer Antifer
cubes with “irregular placement”, 1.25≤cotα≤2.5 and packing densities 1.08≤φ≤1.16
(46%≥p≥42%); the K values obtained by both Yagci and Kapdasli (2003) and Yagci et al.
D
(2004) were lower than common values given in the literature which are based on a higher
recommended packing density φ=1.21 (p=39.5%). After conducting 17 small-scale tests with
double-layer Antifer cubes, cotα=1.5, different placement techniques and armour porosities
0.90≤φ≤1.22 (55%≥p≥39%), Frens (2007) found that higher φ for the same placement
technique, resulted in higher hydraulic stability and high wave reflection, runup and
overtopping rates.
Taking into account the experimental results of the three tests with different armour porosities
and irregular placement reported by Frens (2007), it is clear that an increase of 7% in packing
density (from φ=1.14 to φ=1.22) resulted in a 20% increase in the stability number
corresponding to 10% armour damage. Hydraulic stability of cubes and Antifer cubes clearly
decreases when packing density decreases below the recommended values.

2.4 Double-layer Dolos armours
Carver and Davidson (1978) carried out small-scale tests with double-layer Dolos armours
with randomly-placed units, using regular waves, cotα=1.5 and packing densities
0.63≤φ≤0.83 (68.5%≥p≥58.5%); the armours with a lower packing density also had a lower
hydraulic stability. Zwamborn (1978) conducted similar experiments with higher packing
densities 0.83≤φ≤1.15 (58.5%≥p≥42.5%) finding little difference in damage. Zwamborn
(1978) also recommended φ=1.00 (p=50%) for practical applications, while USACE (1984)
recommended a significantly lower packing density φ=0.83 (p=58.5%).
Burcharth and Liu (1992) tested double-layer Dolos armours with slope cotα=1.5; based on
these results and the observations of other authors, they proposed Eq. 7 with an explicit linear
relationship between packing density and stability number:
H
N = s = (47−72 r)D1/3N−0.1φ (7)
s ∆D
n
in which N is the stability number; r is the Dolos waist ratio; D is the relative number of
s
displaced units; N is the number of waves; and φ is the packing density. For double-layer
Dolos armours, if φ increases, the stability number increases. The Dolos armours showed a
8
linear relationship between N and φ in the range 0.61≤φ≤1.00.
s

2.5 Single-layer AccropodeTM, Core-LocTM and XblocR armours
Several bulky and second-generation armour units (see Dupray and Roberts, 2009) have been
designed to balance hydraulic stability and structural strength for single-layer armouring.
Usually, armour unit placement is a critical issue and both prototype and small-scale models
have to follow specific prescriptions. According to Van Gelderen and Auld (2009),
Accropodes must be placed accurately in a predefined grid so as to obtain the required
interlocking, while Paulsen and Wareing (2009) described in detail ten specific placement
criteria to adequately place Core-LocTM units with a real packing density 0.61≤φ≤0.66
(39.0%≥p≥34.0%). When armour unit interlocking is significant (e.g. AccropodeTM, Core-
LocTM and XblocR), the Hudson formula is no longer valid and steeper slopes (cotα=4/3) are
recommended.
Vincent et al. (1989), Holtzhausen and Zwamborn (1991) and Burcharth et al. (1998)
compared small-scale tests results of single-layer AccropodeTM armours from different
laboratories, and relevant discrepancies were found; single-layer AccropodeTM armour
stability was proved to be highly sensitive to core permeability, while placements were stated
to have been made ‘as prescribed by patent-owner’. However, packing density prescriptions
and recommended placement techniques have been changing over time following engineering
experience and research advances. For instance, Melby and Turk (1994) recommended a
packing density φ=0.54 (p=46.0%) for Core-LocTM armour (slope cotα=4/3), and CLI (2012)
recently recommended design packing densities for AccropodeTM and Core-LocTM depending
on the unit size (φ≈0.65 for the smallest units and φ≈0.62 for the largest units).
Bakker et al. (2005) carried out 2D small-scale tests to analyse the influence of the packing
density on the hydraulic stability of single-layer XblocR armours. A higher packing density,
0.57≤φ≤0.60 (43%≥p≥40%), leads to a higher hydraulic stability; in this range, a 4% increase
in packing density resulted roughly in a 12% increase in the stability number. Furthermore,
models with the lowest packing density, 0.55≤φ≤0.57 (45%≥p≥43%), showed relevant
armour settlements. DMC (2011) recommended a design packing density φ=0.58 (p=42%)
for XblocR armours because Bakker et al. (2005) observed that armours with 0.55≤φ≤0.57
settled to obtain φ≈0.58.
Bulky armour units such as AccropodeTM, Core-LocTM and XblocR, placed in single-layer
armours, must achieve specific armour porosity targets, depending on the geometry and size
9